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Zbl 0974.16007
Crawley-Boevey, William; Holland, Martin P.
Noncommutative deformations of Kleinian singularities. (English)
[J] Duke Math. J. 92, No. 3, 605-635 (1998). ISSN 0012-7094

A Kleinian singularity is the quotient $K^2/\Gamma$, where $K$ is an (algebraically closed) field (of characteristic zero) and $\Gamma$ is a nontrivial finite subgroup of $\text{SL}_2(K)$. More precisely, this is an object whose coordinate ring is $K[x,y]^\Gamma$, where the action of $\Gamma$ on $K[x,y]$ is extended from the given action of $\Gamma$ on the two-dimensional vector space spanned by $x$ and $y$. In the paper under review, the authors define and study a family ${\cal O}^\lambda$ of deformations of $K[x,y]^\Gamma$, where $\lambda\in Z(K\Gamma)$. The definition of ${\cal O}^\lambda$ is as follows. $\Gamma$ acts in an obvious way on the noncommuting polynomials $K\langle x,y\rangle$ and one forms the corresponding skew group ring $K\langle x,y\rangle\Gamma$. For $\lambda\in Z(K\Gamma)$, define ${\cal S}^\lambda$ as the quotient $K\langle x,y\rangle\Gamma/(xy-yx-\lambda)$. Let $e\in K\Gamma$ be the average of the group elements. Then ${\cal O}^\lambda$ is defined as $e{\cal S}^\lambda e$. These rings are Noetherian, finitely generated $K$-algebras, of Gelfand-Kirillov dimension 2. They are also Auslander-Gorenstein and Cohen-Macaulay. Other properties of ${\cal O}^\lambda$ are studied by means of the so called deformed preprojective algebras.\par In a subsequent paper by the second author [Comment. Math. Helv. 74, No. 4, 548-574 (1999; Zbl 0958.16014)], deformed preprojective algebras are embedded in a wider class of algebras, which provides a more conceptual approach to the study of deformations of Kleinian singularities. The reader is referred to that paper for more details.
[Alex Martsinkovsky (Boston)]

MSC 2000:: *16G10 Representations of Artinian rings
14B07 Deformations of singularities (local theory)
16S80 Deformation theory of associative ring and algebras
14A22 Noncommutative algebraic geometry, etc.
16S32 Associative rings of differential operators

Keywords: Kleinian singularities; deformed preprojective algebras; McKay graphs; deformations

Citations: Zbl 0958.16014

Cited in: Zbl 1220.18012 Zbl 1193.16023 Zbl 1138.16012 Zbl 1171.16007 Zbl 1095.15014 Zbl 1153.16011 Zbl 1079.16006 Zbl 1060.16017 Zbl 1059.58006 Zbl 1113.16019 Zbl 1048.14001 Zbl 1036.16024

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